Question

If the equations of the two diameters of a circle are $x+2y=1$ and $x-y=4$ and the radius of the circle is $5$, find the equation of the circle.

Answer 1

Here radius of the circle $=5$

Equations of two diameters of the circle are

$x+2y=1$ ....$\left(1\right)$

$x-y=4$ ....$\left(2\right)$

Solving $\left(1\right)$ and $\left(2\right)$ we get

$\left(2\right)\Rightarrow x=4+y$

Substituting $x=4+y$ in $\left(1\right)$ we get

$4+y+2y=1$

$\Rightarrow 4+3y=1$

$\Rightarrow 3y=1-4=-3$

$\therefore y=-1$

substitute $y=-1$ in $x=4+y$

we get

$x=4-1=3$

Hence centre of the circle is $(3,-1)$

Now the equation of the required circle is

${(x-3)}^2+{(y+1)}^2=5^2$

$\Rightarrow x^2+9-6x+y^2+1+2y-25=0$

$\Rightarrow x^2+y^2-6x+2y-15=0$